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In this paper, we consider a weakly coupled sinh-Gordon equation which takes values in a commutative Frobenius subalgebra of gl(2, C). Then we construct some nonlocal symmetries of the Frobenius sinh-Gordon system using its Bäcklund transformation and infinitesimal transformations. Based on the nonlocal symmetries, we show some conserved densities of the Frobenius sinh-Gordon system. Using these symmetries, we also construct some new coupled integro-differential systems.

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

A hierarchy of integrable lattice equations with three potentials is constructed from a new discrete 3 × 3 matrix spectral problem. It is shown that the hierarchy possesses a Hamiltonian structure and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.

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